A Concentration Behavior for Semilinear Elliptic Systems with Indefinite Weight
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摘要
Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β> 1,α + β< 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n > 0} shrink to a point x_0∈Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n > 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.
Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β> 1,α + β< 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n > 0} shrink to a point x_0∈ Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n > 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.
Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β> 1,α + β< 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n > 0} shrink to a point x_0∈ Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n > 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.
引文
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[5]Bonheure,D.,Gomes,J.M.,Habets,P.:Multiple positive solutions of superlinear elliptic problems with sign-changing weight.J.Differential Equations,214(1),36-64(2005)
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[7]Brown,K.J.,Zhang,Y.:The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function.J.Differential Equations,193,481-499(2003)
[8]Buryak,A.V.,Trapani,P.D.,Skryabin,D.V.,et al.:Optical solitons due to quadratic nonlinearities:from basic physics to futuristic applications.Phys.Rep.,370(2),63-235(2002)
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[12]Girao,P.M-,Gomes,M.J.:Multibump nodal solutions for an indefinite superlinear elliptic problem.J.Differential Equations,247,1001-1012(2009)
[13]Kaminow,I.P.:Polarization in optical fibers.IEEE J.Quantum Electron,17,15-22(1981)
[14]Kartashov,Y.V.,Malomed,B.A.,Torner,L.:Solitons in nonlinear lattices.Rev.Modern.Phys.,83,247-305(2011)
[15]Lopez-Gomez,J.:Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems.Trans.Amer.Math.Soc,352,1825-1858(2000)
[16]Menyuk,C.R.:Nonlinear pulse propagation in birefringence optical fiber.IEEE J.Quantum Electron,23,174-176(1987)
[17]Menyuk,C.R.:Pulse propagation in an elliptically birefringent Kerr medium.IEEE J.Quantum Electron,25,2674-2682(1989)
[18]Pendry,J.B.,Schurig,D.,Smith,D.R.:Controlling electromagnetic fields.Science,312(5781),1780-1782(2006)
[19]Ramos,M.,Terracini,S.,Troestler,C:Superlinear indefinite elliptic problems and Pohozaev type identities.J.Funct.Anal.,159,596-628(1998)
[20]Shalaev,V.M.:Optical negative-indexmetamaterials.Nat.Photon.,1(1),41-48(2007)
[21]Smith,D.R.,Pendry,J.B.,Wiltshire,M.C.K.:Metamaterials and negative refractive index.Science,305(5685),788-792(2004)
[22]Strauss,W.A.:The nonlinear Schroinger equation.Contemporary developments in continuum mechanics and partial differential equations.Proc.Internat.Sympos.,Inst.Mat.,Univ.Fed.Rio de Janeiro,Rio de Janeiro,1977.North-Holland Math.Stud.,Vol.30,North-Holland,Amsterdam,452-465(1978)
[23]Stuart,C.A.:Self-trapping of an electromagnetic field and bifurcation from the essential spectrum.Arch.Ration.Mech.Anal.,113(1),65-96(1991)
[24]Stuart,C.A.:Guidance properties of nonlinear planarwaveguides.Arch.Ration.Mech.Anal.,125(2),145-200(1993)
[25]Stuart,C.A.:Existence and stability of TE modes in a stratified non-linear dielectric.IMA J.Appl.Math.,72(5),659-679(2007)
[26]Szulkin,A.,Willem,M.:Eigenvalue problems with indefinite weight.Studia Math.,135,191-201(1999)
[27]Veselago,V.G.:The electrodynamics of substances with simultaneously negative values of e and fj..PhysicsUspekhi,10(4),509-514(1968)
[28]Wai,P.K.A.,Menyuk.,C.R.,Chen,H.H.:Stability of solitons in randomly varying birefringent fibers.Optim.Lett.,7,40-42(1982)
[2]Akhmediev,N.,Ankiewicz,A.:Partially coherent solitons on a finite background.Phya.Rev.Lett.,82,2661-2664(1999)
[3]Ambrosetti,A.,Arcoya,D.,Gamez,J.L.:Asymmetric bound states of differential equations in nonlinear optics.Rend.Sem.Mat.Univ.Padova,100,231-247(1998)
[4]Berestycki,H.,Capuzzo-Dolcetta,I.,Nirenberg,L.:Variational methods for indefinite superlinear homogeneous elliptic problems.NoDEA Nonlinear Differential Equations Appl.,2,553-572(1995)
[5]Bonheure,D.,Gomes,J.M.,Habets,P.:Multiple positive solutions of superlinear elliptic problems with sign-changing weight.J.Differential Equations,214(1),36-64(2005)
[6]Brown,K.J.,Wu,T.F.:A semilinear elliptic system involving nonlinear boundary condition and signchanging weight function.J.Math.Anal.Appl.,337,1326-1336(2008)
[7]Brown,K.J.,Zhang,Y.:The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function.J.Differential Equations,193,481-499(2003)
[8]Buryak,A.V.,Trapani,P.D.,Skryabin,D.V.,et al.:Optical solitons due to quadratic nonlinearities:from basic physics to futuristic applications.Phys.Rep.,370(2),63-235(2002)
[9]Chabrowski,J.:Weak Convergence Methods for Semilinear Elliptic Equations,World Scientific Publishing Co.Pte.Ltd,Singapore,1999
[10]Dror,N.,Malomed,B.A.:Solitons supported by localized nonlinearities in periodic media.Phys.Rev.A(3),83,033828(2011)
[11]Evangelides,S.G.,Mollenauer,Jr.L.F.,Gordon,J.P.,et al.:Polarization multiplexing with solitons.J.Lightwave Technol.,10,28-35(1992)
[12]Girao,P.M-,Gomes,M.J.:Multibump nodal solutions for an indefinite superlinear elliptic problem.J.Differential Equations,247,1001-1012(2009)
[13]Kaminow,I.P.:Polarization in optical fibers.IEEE J.Quantum Electron,17,15-22(1981)
[14]Kartashov,Y.V.,Malomed,B.A.,Torner,L.:Solitons in nonlinear lattices.Rev.Modern.Phys.,83,247-305(2011)
[15]Lopez-Gomez,J.:Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems.Trans.Amer.Math.Soc,352,1825-1858(2000)
[16]Menyuk,C.R.:Nonlinear pulse propagation in birefringence optical fiber.IEEE J.Quantum Electron,23,174-176(1987)
[17]Menyuk,C.R.:Pulse propagation in an elliptically birefringent Kerr medium.IEEE J.Quantum Electron,25,2674-2682(1989)
[18]Pendry,J.B.,Schurig,D.,Smith,D.R.:Controlling electromagnetic fields.Science,312(5781),1780-1782(2006)
[19]Ramos,M.,Terracini,S.,Troestler,C:Superlinear indefinite elliptic problems and Pohozaev type identities.J.Funct.Anal.,159,596-628(1998)
[20]Shalaev,V.M.:Optical negative-indexmetamaterials.Nat.Photon.,1(1),41-48(2007)
[21]Smith,D.R.,Pendry,J.B.,Wiltshire,M.C.K.:Metamaterials and negative refractive index.Science,305(5685),788-792(2004)
[22]Strauss,W.A.:The nonlinear Schroinger equation.Contemporary developments in continuum mechanics and partial differential equations.Proc.Internat.Sympos.,Inst.Mat.,Univ.Fed.Rio de Janeiro,Rio de Janeiro,1977.North-Holland Math.Stud.,Vol.30,North-Holland,Amsterdam,452-465(1978)
[23]Stuart,C.A.:Self-trapping of an electromagnetic field and bifurcation from the essential spectrum.Arch.Ration.Mech.Anal.,113(1),65-96(1991)
[24]Stuart,C.A.:Guidance properties of nonlinear planarwaveguides.Arch.Ration.Mech.Anal.,125(2),145-200(1993)
[25]Stuart,C.A.:Existence and stability of TE modes in a stratified non-linear dielectric.IMA J.Appl.Math.,72(5),659-679(2007)
[26]Szulkin,A.,Willem,M.:Eigenvalue problems with indefinite weight.Studia Math.,135,191-201(1999)
[27]Veselago,V.G.:The electrodynamics of substances with simultaneously negative values of e and fj..PhysicsUspekhi,10(4),509-514(1968)
[28]Wai,P.K.A.,Menyuk.,C.R.,Chen,H.H.:Stability of solitons in randomly varying birefringent fibers.Optim.Lett.,7,40-42(1982)